On commutative weakly FGI-rings (Q1944279)

Let \(M\) be a unital module over a commutative ring \(R\). Recall that a module \(M\) is called co-Hopfian (resp. weakly co-Hopfian) if any injective endomorphism of \(M\) is an isomorphism (resp. essential). If any co-Hopfian (resp. weakly co-Hopfian) \(R\)-module is finitely generated, then \(R\) is said to be an FGI-ring (resp. weakly FGI-ring). Some properties of weakly FGI-rings are investigated first, for example, it is shown that every prime ideal of a commutative weakly FGI-ring is maximal and the set of all prime ideals of \(R\) is finite. Via a few very technical lemmas it is shown that a weakly co-Hopfian \(R\)-module can be constructed over a commutative Artinian ring \(R\) with a nonprincipal ideal. The main theorem of the paper gives characterizations for a commutative ring \(R\) to be a weakly FGI-ring. To be specific: \(R\) is a weakly FGI-ring if and only if \(R\) is an Artinian principal ideal ring if and only if every weakly co-Hopfian \(R\)-module is quasi-Artinian.